SIGNIFICANT FIGURES
Every numerical measurement contains a certain degree of uncertainty. The extent of the uncertainty depends on the nature of the measuring device and the skill with which it is applied. Significant figures are used by scientists to indicate the degree of uncertainty of numerical measurements.
Consider four measurements and the degree of uncertainty that has been estimated for each measurement.
9 ± 1 cm
9.0 ± 0.1 cm
9.00 ± 0.001 cm
9.000 ± 0.001 cm
These measurements and the degree of uncertainty can be expressed with significant figures as:
9 cm One significant figure
9.0 cm Two significant figures
9.00 cm
9.000 cm
Three significant figures
Four significant figures
The use of significant figures indicates to the knowledgeable reader that there is uncertainty of at least one unit in the last digit. If there is a zero as the last digit to the right of the decimal pint, the zero is assumed to be significant. Therefore, 1762.0 has five significant figures. However, the number 1500 has only two significant figures, while the number 1501 has four. Writing 1500 in significant figures unambiguously indicates the number of significant figures. 1.5000 x 10
3
has five significant figures, 1.5 x 10
3
has two.
RULES FOR SIGNIFICANT FIGURES
Rule Number
A number is a significant figure if it is: a. not a zero b. a zero between nonzero digits c. a zero on either side of a decimal point
4.5 g
122.35 m
205 m
5.082 kg
50.0 L
25.0 o C
16.00 g
0.00230
4.0 x 10 5 m
5.7000 x10 -3 kg
3.02 x 10 4 years
Significant figures
2
5
3
4
3
3
4
3
2
5
3 d. any digit in the coefficient of a number written in scientific notation e. the number of significant figures is determined starting with the leftmost non-zero digit. The leftmost non-zero digit is the most significant digit or figure. f. the rightmost digit of a decimal number is the least significant digit or figure because it has the greatest degree of uncertainty.
A zero is not significant if it is: a. at the beginning of a decimal number b. used as a placeholder in a large number without a decimal point
0.0004 kg
0.075 km
50 kg
850000 km
1250000 years
1
2
3
1
2

Rules for Rounding off
1. If the first digit to be dropped is 4 or less, it and all following digits are dropped from the number.
Examples: 4.501 x 10
3
years in two significant numbers is 4.5 x 10
3
years.
43.21 g expressed in three significant numbers is 43.2 g.
453.46 km expressed in three significant numbers is 453 km.
456.36 km expressed in four significant numbers is 456.3 km
2. If the first number to be dropped is 5 or greater, the last retained digit of the number is increased by 1.
Examples: 4.55 x 10
3
years in two significant numbers is 4.6 x 10
3
years.
456.79 km expressed in two significant numbers is 460 km.
43.27 g expressed in three significant numbers is 43.3 g.
456.79 km expressed in four significant numbers is 456.8 km.
Some numbers are a little tricky to round. Consider the measurement 1845.449 km.
Significant figures
Number
1
2000
2
1800
3
1850
4
1845
5
1845.4
6
1845.45
7
1845.449
Rules for Significant Figures in Calculations
The number of significant figures used limits the number of figures reported after a number is calculated by subtraction, multiplication, and division or any other mathematical operation.
Addition and Subtraction
The uncertainty is determined by the absolute uncertainty in the least precise measurement, not by the number of significant figures. For the purposes of this course, this is the number of digits after the decimal point.
Examples: 32.01 km + 5.325 km + 12 km = 49.335 km = 49 km
32.010 km + 5.325 km + 12.00 km = 49.335 km = 49.34 km
32.0100 km + 5.325 km + 12.000 km = 49.335 km = 49.335 km
14.5 - 2.5 = 12 = 12.0
153.247 – 14.82 = 138.43
Multiplication and Division
For multiplication and division, the final answer is written so it has the same number of significant figures as the measurement with the fewest significant figures.
Examples: 32.01 km x 5.325 km x12 km = 2045.439 km = 2000 km
32.010 km x 5.325 km x 12.00 km = 2045.439 km = 2045 km
32.0100 km x 5.3250 km x 12.000 km = 2045.439 km = 2045.4 km
56.8 x 0.37 = 21.016 = 21
71.4/11 = 6.4909091 = 6.5
25.0/5.00 = 5.00